ch17

Advanced Engineering Mathematics

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CHAPTER 17 Conformal Mapping This is a new chapter. It collects and extends the material on conformal mapping contained in Chap. 12 of the previous edition. SECTION 17.1. Geometry of Analytic Functions: Conformal Mapping, page 729 Purpose. To show conformality (preservation of angles in size and sense) of the mapping by an analytic function w 5 ƒ( z ); exceptional are points with ƒ 9 ( z ) 5 0. Main Content, Important Concepts Concept of mapping. Surjective, injective, bijective Conformal mapping (Theorem 1) Magnification, Jacobian Examples. Joukowski airfoil Comment on the Proof of Theorem 1 The crucial point is to show that w 5 ƒ( z ) rotates all straight lines (hence all tangents) passing through a point z 0 through the same angle a 5 arg ƒ 9 ( z 0 ), but this follows from (3). This in a nutshell is the proof, once the stage has been set. SOLUTIONS TO PROBLEM SET 17.1, page 733 2. By conformality 4. u 5 x 2 2 y 2 , v 5 2 xy , y 2 5 v 2 /(4 x 2 ), x 2 5 v 2 /(4 y 2 ); hence for x 5 1, 2, 3, 4 we obtain u 5 x 2 2 v 2 /(4 x 2 ) 5 1 2 v 2 /4, 4 2 v 2 /16, 9 2 v 2 /36, 16 2 v 2 /64 and for y 5 1, 2, 3, 4, u 5 v 2 /(4 y 2 ) 2 y 2 5 v 2 /4 2 1, v 2 /16 2 4, v 2 /36 2 9, v 2 /64 2 16. 6. w 5 1/ z is called reflection in the unit circle. The answer is u w u 5 3, 2, 1, 1/2, 1/3 and Arg w 5 0, 7 p /4, 7 /2, 7 3 /4, 7 . We see that the unit circle is mapped onto itself, but only 1 and 2 1 are mapped onto themselves. e 2 i u is mapped onto e 2 i , the complex conjugate, for example, i onto 2 i . 8. On the line x 5 1 we have z 5 1 1 iy , w 5 u 1 i v 5 1/ z 5 (1 2 iy )/(1 1 y 2 ), so that v 52 y /(1 1 y 2 ) and, furthermore, u 5 , u (1 2 u ) 5 ( 1 2 ) 55 v 2 . Having obtained a relation between u and v , we have solved the problem. We now have u 2 2 u 1 v 2 5 0, ( u 2 1 _ 2 ) 2 1 v 2 5 1 _ 4 and by taking roots u w 2 1 _ 2 u 5 1 _ 2 , a circle through 0 and 1 whose interior is the image of x . 1. 10. The lower half-plane y 2 ±± (1 1 y 2 ) 2 1 ± 1 1 y 2 1 ± 1 1 y 2 1 ± 1 1 y 2 280 im17.qxd 9/21/05 1:08 PM Page 280
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12. Annulus 1/ e , u w u , e cut along the negative real axis 14. Whole w -plane except w 5 0 16. CAS Experiment. Orthogonality is a consequence of conformality because in the w -plane, u 5 const and v 5 const are orthogonal. We obtain (a) u 5 x 4 2 6 x 2 y 2 1 y 4 , v 5 4 x 3 y 2 4 xy 3 (b) u 5 x /( x 2 1 y 2 ), v 52 y /( x 2 1 y 2 ) (c) u 5 ( x 2 2 y 2 )/( x 2 1 y 2 ) 2 , v 2 xy /( x 2 1 y 2 ) 2 (d) u 5 2 x /((1 2 y ) 2 1 x 2 ), v 5 (1 2 x 2 2 y 2 )/((1 2 y 2 ) 2 1 x 2 ) 18. 2 z 2 2 z 2 3 5 0, z 4 5 1, hence 6 1, 6 i . 20. (cosh 2 z ) 9 5 2 sinh 2 z 5 0 at z 5 0, 6 p i /2, 6 i , ••• 22. (5 z 4 2 80) exp ( z 5 2 80 z ) 5 0, z 4 5 16, z 56 2, 6 2 i 24. M 5 u z u 5 1 on the unit circle, J 5 u z u 2 26. M 5 3 u z u 2 5 1 on the circle u z u 5 1/ Ï 3 w , J 5 9 u z u 4 28. w 9 52 1/ z 2 , M 5 u w 9 u 5 1/ u z u 2 5 1 on the unit circle, J 5 1/ u z u 4 30. By the Taylor series, since the first few derivatives vanish at z 0 , ƒ( z ) 5 ƒ( z 0 ) 1 ( z 2 z 0 ) k g ( z ), g ( z 0 ) ± 0, since ƒ ( k ) ( z 0 ) ± 0.
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ch17 - im17.qxd 1:08 PM Page 280 CHAPTER 17 Conformal...

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